[{_id:1,objects:[{prop_1:"...",foo:"a"},{prop_1:"...",foo:"a"},{prop_1:"..."}]},{_id:2,objects:[{prop_1:"...",foo:"a"},{prop_1:"...",foo:"a"},{prop_1:"..."}]},{_id:3,objects:[{prop_1:"...",foo:"a"},{prop_1:"...",foo:"a"},{prop_1:"...",foo:"a"}]}]

db.collection.aggregate([{$match:{$expr:{$setIsSubset:[{$map:{input:"$objects",in:"$$this.foo"}},["a"]]}}}])